This is a very interesting premise, though I don't know if it would be an absolute career killer for Nixon. From what I recall, Eisenhower also considered picking Massachusetts Governor Christian Herter for the VP spot in '56 because he wasn't as universally hated by Democrats on the hill. Let's say that Ike picks Herter and Nixon goes to Secretary of Defense.
Herter probably won't run in 1960, because for the most part, the Vice Presidency wasn't considered the POTUS' direct successor until Nixon built the office up and ran of his own accord in 1960. Without a strong candidate announcing right out of the gate, Rocky and Goldwater announce as well, and Nixon makes his bid from the center from within the Pentagon. He probably wins the nomination, but may lose the general to Kennedy, who could blame Nixon for the imaginary 'missile gap' that helped him clench the election IOTL.
After that, maybe Nixon wins the Governorship in 1962 thanks to butterflies? His career could turnout the same way, really, though he'd be much more learned on military issues and management than he was IOTL. How might that effect a Nixon Presidency in the late sixties and early seventies, especially with regard to Vietnam? Might Nixon move to more strongly escalate the war or change gears completely?